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Question 1: Will a comet’s orbit by faster at perihelion or aphelion?        Question 2: The semi-major axis of Halley’s Comet is 17.8 AU. How long does it take Halley’s Comet to complete one orbit around the Sun

Question 1: Will a comet’s orbit by faster at perihelion or aphelion?        Question 2: The semi-major axis of Halley’s Comet is 17.8 AU. How long does it take Halley’s Comet to complete one orbit around the Sun

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pic comes first, and is this possible to do?

In the above graphic, b is the length of one side of the triangle, here defined as the “base”. If you drop a perpendicular from the angle opposite side b to side b, this is called the “height of the triangle”. The area of the triangle, called A, is related to the base and height using the equation: 

 

                                           A = (1/2) (base X height) = 0.5bh. 

 

In the case where b = 10 cm and h =6 cm, A = 0.5 X 10 X 6 = 30. 

 

Kepler’s Three Laws of Elliptical Planetary Motion are written: 

 

  1. Planets move in elliptical orbits around the Sun. 
  1. The line connecting a planet with the Sun, called the Radius Vector, sweeps out equal areas in equal times. 
  1. The Orbital Period of a planet around the Sun, P, is the time it takes that planet to complete one solar orbit. The square of P is equal is proportional to the planet’s semi-major axis, a.  This is expressed using the equation: 

 

                                               P2 = Constant a3. 

 

      If the planet’s orbital period is the years (yr) and the planet’s semi-major axis is in 

      Astronomical Units (au), the Constant in the above equation is equal to 1 and the 

      equation becomes: 

 

                                                 Pyr2 =  aau3. 

Kepler’s Second Law: Radius Vector Sweeps Out Equal Areas in Equal Times: 

2ND pic

 

 Table of Data for Kepler’s Second Law 

 

Triangle Number   Base Length (b)   Height Length (h)  A = Triangle Area ( bh/2)     PE 

           1 

           2 

           3 

           4 

           5  

           6 

           7 

           8 

 ___________________________________________________________________  

                                                                                   Average Area (Aav) 

           

Please calculate average area by adding together all of the triangle areas and dividing by 8.  To calculate Percent Error (PE)  for each triangle area, use 

 

                          (A - Aav) 

PE  =         100  ________ , where A is the area of each triangle. 

                             Aav  

Kepler’s Third Law: Pyr2 =  aau3. 

                      

The table below presents the semi-major axis (a) and Actual orbital period for all of the major planet’s in the solar system. Cube for each planet the semi-major axis in Astronomical Units. Then take the square root of this number to get the Calculated orbital period of each planet. Fill in the final row of data for each planet. 

 

                            Table of Data for Kepler’s Third Law: 

 

Planet              aau = Semi-Major Axis (AU)   Actual Planet      Calculated Planet  

                                                                        Period (Yr)            Period (Yr) 

__________   ______________________   ___________    ________________ 

Mercury                      0.39                                0.24 

Venus                         0.72                                0.62 

Earth                          1.00                                1.00 

Mars                           1.52                                1.88 

Jupiter                        5.20                              11.19 

Saturn                        9.54                              29.50 

Uranus                     19.20                              84.00 

Neptune                   30.10                            164.80 

__________________________________________________________________  

 

 

Question 1: Will a comet’s orbit by faster at perihelion or aphelion? 

 

 

 

Question 2: The semi-major axis of Halley’s Comet is 17.8 AU. How long does it take Halley’s Comet to complete one orbit around the Sun? 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here is a scanned version of a reduced centimeter ruler. You may  choose to use you own. 

The average distance of the Earth from the Sun is 150 million kilometers (1.5 X 108 km). This distance is defined as į Astronomical Unit or 1 AU. Mercury, the closest of the major planets to the Sun, is about 0.4 AU from the Sun. Neptune, the farthest known major planet from the Sun is at about 30 AU from the Sun. All of the major planets and most asteroids are in nearly circular solar orbits. Many comets, on the other hand, are in highly elliptical solar orbits. One of the essential mathematical taols in considering elliptical solar orbits is the calculation of the area of a triangle. AREA OF TRIANGLE B BYJU'S The Learing App Base Area =x base x perpendicular height O Byjus.com Height
Transcribed Image Text:The average distance of the Earth from the Sun is 150 million kilometers (1.5 X 108 km). This distance is defined as į Astronomical Unit or 1 AU. Mercury, the closest of the major planets to the Sun, is about 0.4 AU from the Sun. Neptune, the farthest known major planet from the Sun is at about 30 AU from the Sun. All of the major planets and most asteroids are in nearly circular solar orbits. Many comets, on the other hand, are in highly elliptical solar orbits. One of the essential mathematical taols in considering elliptical solar orbits is the calculation of the area of a triangle. AREA OF TRIANGLE B BYJU'S The Learing App Base Area =x base x perpendicular height O Byjus.com Height
The graphic on the next page refers to Kepler's 2nd Law for the elliptical solar orbit of a hypothetical comet. The numbers on the orbit (0-7) refer to time. The time interval between points 0 and 1 is the same as that beiween points 1 and 2, betwveen points 2 and 3, between points 3 and 4, between points 4 and 5, between points 5 and between points 6 and 7 and points 7 and'o. Each arc between each pair of points on the orbit and the Sun define a triangle. In the table below the graphic. Triangle 1 is defined by points o and 1 on the orbit and the Sun. Triangle 8 is defined by points 7 and O on the orbit and the Sun. You will calculate the areas of each triangle as discussed in the Theory section above. For Triangle 1. take the base as the line between the Sun and point O'on the orbit. For Triangle 2, take the base as the line between the Sun and point 1 on the orbit. Construct the Height h for each triangle. Measure the length in centimeters of h and b for each triangle as accurately as you can using the scanned centimeter ruler on p. 17 Or your own centimeter ruler. Calculate all area and enter them in the table below Keplers 2nd Law
Transcribed Image Text:The graphic on the next page refers to Kepler's 2nd Law for the elliptical solar orbit of a hypothetical comet. The numbers on the orbit (0-7) refer to time. The time interval between points 0 and 1 is the same as that beiween points 1 and 2, betwveen points 2 and 3, between points 3 and 4, between points 4 and 5, between points 5 and between points 6 and 7 and points 7 and'o. Each arc between each pair of points on the orbit and the Sun define a triangle. In the table below the graphic. Triangle 1 is defined by points o and 1 on the orbit and the Sun. Triangle 8 is defined by points 7 and O on the orbit and the Sun. You will calculate the areas of each triangle as discussed in the Theory section above. For Triangle 1. take the base as the line between the Sun and point O'on the orbit. For Triangle 2, take the base as the line between the Sun and point 1 on the orbit. Construct the Height h for each triangle. Measure the length in centimeters of h and b for each triangle as accurately as you can using the scanned centimeter ruler on p. 17 Or your own centimeter ruler. Calculate all area and enter them in the table below Keplers 2nd Law
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